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Relative Aging of Distributions

Published online by Cambridge University Press:  27 July 2009

Ginger Rowell  and
Kyle Siegrist
Ginger Rowell
Affiliation:
Department of Mathematics and Computer Science, Belmont University, Nashville, Tennessee 37212, [email protected]
Kyle Siegrist
Affiliation:
Department of Mathematical Sciences, University of Alabama in Huntsville, Huntsville, Alabama [email protected]
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Abstract

We consider the reliability of one random variable relative to another, when the variables are continuous and take values in an interval [a, b). We give definitions and characterizations for the exponential property and the standard aging properties IFR, IFRA, and NBU. The exponential property defines an equivalence relation on the distributions, and then each of these aging properties defines a partial order on the distributions, modulo the exponential equivalence. We give a set of conditions that must be satisfied for a general aging property to define such a partial order and show that these conditions are not satisfied by the NBUE property. Several parametric families of distributions are considered.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 12 , Issue 4 , October 1998 , pp. 469 - 478
Copyright
Copyright © Cambridge University Press 1998

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References

1.Barlow, R.E. & Proschan, F. (1965). Mathematical theory of reliability. New York: John Wiley and Sons.Google Scholar
2.Barlow, R.E. & Proschan, F. (1966). Inequalities for linear combinations of order statistics from restricted families. Annals of Mathematical Statistics 37: 1574–1592.CrossRefGoogle Scholar
3.Barlow, R.E. & Proschan, F. (1975). Statistical theory of reliability and life testing. Atlanta: Holt Rinehart and Winston.Google Scholar
4.Gertsbakh, I.B. (1989). Statistical reliability theory. New York: Marcel Dekker.Google Scholar
5.Rowell, G.H. (1995). Probability distributions on temporal semigroups. Ph.D. dissertation, University of Alabama in Hunts ville.Google Scholar
6.Shaked, M. & Shanthikumar, J.G. (1994). Stochastic orders and their applications. San Diego: Academic Press.Google Scholar
7.Siegrist, K.T. (1994). Exponential distributions on semigroups. Journal of Theoretical Probability 7: 725–737.CrossRefGoogle Scholar